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Asymptotic evolution of certain observables and completeness in Coulomb scattering. I. (English) Zbl 0531.47008
Like the quantum mechanical variables position Q and momentum P, the generator A of the dilation group in $$L^ 2({\mathbb{R}}^{\nu})$$ enjoys a conjugate relation with log $$H_ 0$$, where $$H_ 0$$ is the free non- relativistic Schrödinger operator. This recognition leads to a remarkably simple Cook-type proof for the existence and completeness of two body short-range scattering [V. Enss Commun. Math. Phys. 61, 285-291 (1978; Zbl 0389.47005), E. Mourre Commun. Math. Phys. 68, 91-94 (1979; Zbl 0429.47006), K. B. Sinha, A simple proof of asymptotic completeness in scattering, I.S.I. report (unpublished) (1981)]. For long range potentials, one needs to study the asymptotic details of the scaled observable A/t under the total evolution $$V_ t\equiv \exp(-iHt)$$. In the present paper, the authors study the zeroth order asymptotics and observe that $$V^*_{\tau}\frac{A}{t}V_ t$$ behaves like $$H_ c$$, the continuous part of H. This alone is sufficient to prove completeness in the case of Coulomb potential [see also V. Enss Commun. Math. Phys. 89, 245-268 (1983)]. However, for other longer rage potentials, one needs higher order asymptotics of $$V^*_{\tau}\frac{A}{t}V_ t$$ near $$t=\pm \infty$$ and this was done by the authors in [Asymptotic completeness in long range scattering II, to appear in Ann. Sci. Ecole Norm. Sup. 17, No.4 (1984)].

##### MSC:
 47A40 Scattering theory of linear operators 47A20 Dilations, extensions, compressions of linear operators 81U05 $$2$$-body potential quantum scattering theory
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##### References:
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